====== Logistic Equation ======

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The logistic equation, originally published by [[wp>Pierre François Verhulst]] and often called the Verhulst equation, is often used to describe the dynamics of populations.  It derives from the simplest type of interactions within a species of undifferentiated individuals, with spontaneous (asexual) birth and pairwise competition:

\[
\begin{array}{rcll}
  N   & \xrightarrow{r}      & 2 N & \text{(birth)} \\
  2 N & \xrightarrow{\delta} & N   & \text{(competition).}
\end{array}
\]

Under the assumptions of reaction kinetics (eg. large, well-mixed population) the density of individuals, \( n = [N] \), obeys the [[wp>logistic equation]]:
\[
  \frac{dn}{dt} = r n \left( 1 - \frac{n}{K} \right)
\]
where \( K=r/\delta \).